New existence results on periodic solutions of non-autonomous second order Hamiltonian systems

In this paper, we are concerned with the existence of periodic solutions for the following non-autonomous second order Hamiltonian systems ü(t)+∇F(t,u(t))=0, a.e. t∈[0,T],u(0)−u(T)=u̇(0)−u̇(T)=0,where F:R×RN→R is T-periodic (T>0) in its first variable for all x∈RN. When potential function F(t,x) is either locally in t asymptotically quadratic or locally in t superquadratic, we show that the above mentioned problem possesses at least one T-periodic solutions via the minimax methods in critical point theory, specially, a new saddle point theorem which is introduced in Schechter (1998).